Transcript
Page 1: Knightian uncertainty and moral hazardcshannon/wp/agent-jet.pdfKnightian uncertainty and moral hazard ... Ben Polak, the associate editor, and referees for thoughtful comments. * Corresponding

Journal of Economic Theory 146 (2011) 1148–1172

www.elsevier.com/locate/jet

Knightian uncertainty and moral hazard ✩

Giuseppe Lopomo a, Luca Rigotti b,∗, Chris Shannon c

a Fuqua School of Business, Duke University, United Statesb Department of Economics, University of Pittsburgh, United States

c Department of Economics, University of California, Berkeley, United States

Received 12 November 2009; final version received 30 January 2011; accepted 7 March 2011

Available online 29 March 2011

Abstract

This paper presents a principal-agent model in which the agent has imprecise beliefs. We model thissituation formally by assuming the agent’s preferences are incomplete as in Bewley (1986) [2]. In thissetting, incentives must be robust to Knightian uncertainty. We study the implications of robustness for theform of the resulting optimal contracts. We give conditions under which there is a unique optimal contract,and show that it must have a simple flat payment plus bonus structure. That is, output levels are divided intotwo sets, and the optimal contract pays the same wage for all output levels in each set. We derive this resultfor the case in which the agent’s utility function is linear and then show it also holds if this utility functionhas some limited curvature.© 2011 Elsevier Inc. All rights reserved.

JEL classification: D82; D81; D23

Keywords: Knightian uncertainty; Moral hazard; Contract theory; Incomplete preferences

1. Introduction

In this paper, we study a principal-agent model in which the agent has imprecise beliefs.Our model is motivated by situations in which the agent is less familiar with the details of theproduction process than the principal. For example, imagine an agent about to start a new job.Output depends on his effort as well as variables beyond his control, such as the work of others.

✩ We are grateful to Truman Bewley, Ben Polak, the associate editor, and referees for thoughtful comments.* Corresponding author.

E-mail addresses: [email protected] (G. Lopomo), [email protected] (L. Rigotti), [email protected](C. Shannon).

0022-0531/$ – see front matter © 2011 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2011.03.018

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1149

Consequently he cannot precisely evaluate the stochastic relation between his effort and output.In contrast, the principal owns the production technology and can evaluate the relation betweeneffort and output more precisely. In other words, the principal and the agent have different con-fidence in their beliefs.

The standard literature neglects this possibility by assuming that both principal and agentevaluate state-contingent contracts using expected utility, and hence both precisely evaluate thestochastic consequences of the agent’s action. Asymmetric confidence in beliefs, on the otherhand, is easily described using the Knightian uncertainty theory developed in [2], which providesfoundations for a model in which beliefs are given by sets of probability distributions. Thus boththe possibility of imprecise beliefs and a ranking in which the agent’s beliefs are less precise thanthe principal’s can be formalized in this framework.

Our main result characterizes optimal contracts in a principal-agent model with Knightianuncertainty where the parties have asymmetric confidence in beliefs and linear utility. We giveconditions under which there is a unique optimal contract, and show that it has a simple flatpayment plus bonus structure. In particular, a contract is optimal only if it takes two values acrossall states. The unique optimal incentive scheme divides all possible output levels into two groups,and pays a fixed wage within each group. This result extends to the case in which the agent’sutility function has a limited amount of curvature. This feature of optimal contracts is strikingbecause standard moral hazard models that rule out Knightian uncertainty typically generatecomplex incentive structures. Optimal contracts often involve as many different payments asthere are possible levels of output.

Some authors have speculated that contracts are simple because they need to be robust. Hartand Holmstrom [12], for example, argue that real world incentives need to perform well acrossa wide range of circumstances. Once this need for robustness is considered, simple optimalschemes might obtain. Our results can be viewed through this lens since we introduce robustnessto Knightian uncertainty. In our model this robustness, when paired with asymmetric confidencein beliefs, forces incentive schemes to be simple. Furthermore, these contracts have a shape wecommonly observe since they can be written as a flat payment plus bonus.1

In the spirit of [2], we assume the agent’s preferences are not necessarily complete; he maybe unable to rank all pairs of contracts offered to him. As in the model axiomatized in [2], weassume that these preferences are represented by a utility function and a set of probability dis-tributions, and that the agent prefers one contract to another if the former has higher expectedutility for every probability distribution in this set. Bewley [2] argues that this approach formal-izes the distinction between risk and uncertainty introduced in [16]. In this framework a uniqueprobability is appropriate only when the decision maker regards all events as risky; the decisionmaker uses a set of distributions when he regards some events as uncertain.

With incomplete preferences, choices may be indeterminate and incentive constraints hardto satisfy.2 Bewley [2] proposes a behavioral assumption, inertia, that sometimes alleviates thisproblem. The inertia assumption states that, when faced with incomparable options, an individual

1 Holmstrom and Milgrom [14] provide conditions under which linear incentive schemes are optimal. These conditionsinclude constant relative risk aversion and a specific dynamic property of stochastic output. Neither of these requirementsis related to the idea of robustness considered here. We allow the agent to consider many stochastic structures of output.We also adopt a different notion of simplicity. A two-wage scheme is simple because it can be thought of as contingenton only two events; a linear contract is simple because it is contingent on an intercept and a slope for all events.

2 In a related paper, [17], we study general mechanism design problems. We show that in many standard mechanismdesign settings interim incentive compatibility is equivalent to ex post incentive compatibility.

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1150 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

chooses a status quo or reference point, often interpreted as his current behavior, unless there isan alternative that is strictly preferred. In the standard moral hazard setting, there is a naturalcandidate for the status quo: the agent’s outside option or reservation utility. We consider twocorresponding notions of implementation, depending on whether inertia is incorporated. Withoutinertia, an incentive scheme implements a particular action if the agent prefers that action tohis reservation utility and to all other actions. With inertia, an incentive scheme implements anaction if this action is preferred to the reservation utility and is preferred to all other actions thatare comparable to the reservation utility. With inertia, implementing an action is easier becausethe desired action may be incomparable to some actions. We show that optimal contracts musttake two values under either notion of implementation. Thus while inertia might change the costof implementing an action, it does not change our main result regarding the nature of optimalcontracts.

Our moral hazard model has standard features: the principal cannot observe the agent’s actionand looks for contracts that implement each action at the lowest possible expected cost; eachaction has a different disutility to the agent; each action induces different beliefs over outputoutcomes. We initially assume that principal and agent have linear utility functions over money,and then extend the main result to the case in which the agent’s utility function is not necessarilylinear. The agent perceives Knightian uncertainty, and has beliefs described by sets of probabilitydistributions, one set for each action, while the principal’s beliefs are a unique element of therelative interior of those sets. This formalizes the idea that the agent’s beliefs are more imprecisethan the principal’s. Thus multiplicity of the agent’s beliefs is the only formal difference betweenour model and a linear utility version of [11].

Our model shares important features with standard moral hazard models with both risk-averseand risk-neutral agents. As in models with risk-neutral agents, the agent is indifferent over purelyobjective mean-preserving randomizations that are risky but not uncertain. As in models withrisk-averse agents, however, the agent cares about the distribution of payoffs across subjectivelyuncertain states. We show that optimal contracts with Knightian uncertainty can differ from thosewithout. For example, unlike the standard risk-neutral model, it is no longer possible for theprincipal to do away with incentive concerns by selling the firm to the agent. Thus Knightianuncertainty provides an alternative to limited liability as an explanation for the importance ofincentive provision in many settings where agents are plausibly indifferent to objective mean-preserving lotteries over money.3 Unlike the standard risk-averse model, optimal incentives havea coarse structure regardless of other fine details of the model, including agents’ beliefs.

Mukerji [18] and Ghirardato [9] present moral hazard models similar in motivation to theone presented here. They use a different model of ambiguity, and in contrast show that incen-tive schemes are similar to those in a standard model. In both cases, ambiguity is modeled byassuming that the principal and the agent are Choquet expected utility maximizers with con-vex capacities (see [22]). Neither examines the role of asymmetric confidence, as in each modelthe principal and the agent share the same belief set. On the other hand, Mukerji [19] uses thisframework to relate uncertainty to contract incompleteness.

The paper is organized as follows. The next section briefly describes the model of decisionmakers with incomplete preferences. Section 3 presents the basic framework and discusses theimplementation rules. Section 4 characterizes optimal incentive schemes. Section 5 introduces aprimitive model and establishes versions of our main results in this setting. Section 6 concludes.

3 We thank a referee for suggesting and stressing the importance of this point.

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2. Incomplete preferences and inertia

von Neumann and Morgenstern were the first to observe that completeness is not a satisfac-tory axiom for choice under uncertainty.4 This idea was pursued by Aumann [1] who proposesa preference representation theorem for incomplete preferences and objective probabilities.5 Ina series of papers, [2–4], Bewley develops Knightian decision theory, a model which allows forsubjective probabilities and incompleteness.6

2.1. Incomplete preferences

Under completeness, any pair of alternatives can be ranked. If preferences are not complete,some alternatives are incomparable. Bewley [2] axiomatizes a model allowing for incomplete-ness with subjective probabilities. To formalize this discussion, let the state space N be finite,and index the states by i = 1, . . . ,N . �(N ) := {π ∈ RN : πi � 0 ∀i,

∑i πi = 1} denotes the set

of probability distributions over N , and x, y ∈ XN are random monetary payoffs where X ⊂ Ris finite. In an Anscombe–Aumann setting, Bewley [2] characterizes incomplete preference re-lations represented by a unique nonempty, closed, convex set of probability distributions Π anda continuous, strictly increasing, concave function u :X → R, unique up to positive affine trans-formations, such that

x � y if and only ifN∑

i=1

πiu(xi) >

N∑i=1

πiu(yi) for all π ∈ Π

With a small abuse of notation, we can rewrite this as

x � y if and only if Eπ

[u(x)

]> Eπ

[u(y)

]for all π ∈ Π

where Eπ [·] denotes the expected value with respect to the probability distribution π , and u(x)

denotes the vector (u(x1), . . . , u(xN)).7

The set of probabilities Π reduces to a singleton whenever the preference ordering is com-plete, in which case the usual expected utility representation obtains. Without completeness,comparisons between alternatives are carried out “one probability distribution at a time”, withone bundle preferred to another if and only if it is preferred under every probability distributionconsidered by the agent.8 Bewley [2] suggests that this approach formalizes the distinction be-tween risk and uncertainty originating in [16]. Informally, the size of Π measures how muchuncertainty the individual perceives, and can be thought of as reflecting confidence in beliefs.

Fig. 1 illustrates Bewley’s representation for the special case in which u is linear. The axesmeasure utility (or money) in each of the two possible states. Given a probability distribution,

4 They write:

“It is conceivable – and may even in a way be more realistic – to allow for cases where the individual is neither ableto state which of two alternatives he prefers nor that they are equally desirable... How real this possibility is, both forindividuals and for organizations, seems to be an extremely interesting question... It certainly deserves further study.”[24, Section 3.3.4, p. 19].

5 Aumann’s work has been extended and clarified by [8] and [23].6 [2] has been published recently as [5].7 This representation has been extended by Ghirardato et al. [10] in a Savage setting.8 The natural notion of indifference in this setting says two bundles are indifferent whenever they have the same

expected utility for each probability distribution in Π .

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1152 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

Fig. 1. Incomplete preferences.

a line through the bundle x represents all the bundles that have the same expected value as x

according to this distribution. As the probability distribution changes, we obtain a family of theseindifference curves representing different expected utilities according to different probabilities.The thick curves represent the most extreme elements of this family, while thin curves representother possible elements.

In Fig. 1, y is preferred to x since it lies above all of the indifference curves corresponding tosome expected utility of x. Also, x is preferred to w since w lies below all of the indifferencecurves through x. Finally, z is not comparable to x since it lies above some indifference curvesthrough x and below others. Incompleteness induces three regions: bundles preferred to x, dom-inated by x, and incomparable to x. This last area is empty only if there is a unique probabilitydistribution over the two states and the preferences are complete.

For any bundle x, the better-than-x set has a kink at x. This kink is a direct consequenceof the multiplicity of probability distributions in Π , and vanishes only when Π is a singleton.As a consequence, the assumption of linear utility generates different behavior with multipleprobabilities than in the standard expected utility model. Under expected utility, a linear utilityindex implies a constant marginal rate of substitution and indifference to mixtures of bundles ina given indifference class. Neither of these obtains when there is Knightian uncertainty. This caneasily be seen in Fig. 1, where convex combination of bundles that are not comparable to x cangive rise to a bundle that is strictly preferred to x.

2.2. The inertia assumption

Revealed preference arguments must take incompleteness into account. If x is chosen when y

is available, one cannot conclude that x is revealed preferred to y, but only that y is not revealedpreferred to x. The concepts of status quo and inertia introduced in [2] can sharpen revealedpreference inferences when preferences are incomplete. Bewley’s inertia assumption posits theexistence of planned behavior that is taken as a reference point, and assumes that this “status quo”is abandoned only for alternatives preferred to it. In Fig. 1, for example, if x is the status quo andthe inertia assumption holds, alternatives like z will not be chosen since they are incomparableto x.

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In many economic contexts, there may not be a natural status quo. In the moral hazard modelthat follows, however, an obvious candidate for the status quo is the action yielding the agent’sreservation utility. This corresponds to the payoff the agent receives if he rejects the contractoffered by the principal.

3. The moral hazard setup

The main features of our model are standard and follow the parametrized distribution ap-proach: the principal owns resources that yield output, but needs the agent’s input for productionto take place; principal and agent observe realized output, but the principal does not observe theagent’s action; each action imposes a cost on the agent and generates beliefs about output; thesebeliefs are common knowledge. The main innovation is that beliefs reflect Knightian uncertaintyabout output levels. The crucial assumption is that the agent perceives more uncertainty thanthe principal, and we simplify matters by assuming that the principal perceives no Knightianuncertainty.

Output is an N -vector y = (y1, . . . , yN), with states labeled so that yN > · · · > y1 (throughout,we use subscripts to denote states). A contract is an N -vector w = (w1, . . . ,wN), where wj isthe payment from the principal to the agent in state j . The agent’s reservation utility is w; wewill interpret it as the status quo when we impose the inertia assumption. The agent chooses anaction a from the finite set A := {1,2, . . . ,M}. The principal’s beliefs are described by a functionπP : A → �, while the agent’s beliefs are described by a correspondence Π : A → 2�, whereΠ(a) is nonempty, closed and convex for each a. In the parametrized distribution formulationof the principal-agent problem pioneered by Holmstrom [13], Π is a function and is identicalto πP .9

We assume that beliefs are consistent, but reflect different degrees of precision. Formally, weassume that for each a in A, πP (a) ∈ Π(a). Uniqueness of the principal’s beliefs is assumed foranalytical tractability. Most of the analysis carries over if πP is also a set, as long as asymmetricconfidence holds (i.e. πP (a) is a proper subset of Π(a) for each a).

The agent’s disutility of action a is denoted c(a). Actions are ordered such that whenevera > a′, c(a) > c(a′) and

∑Ni=1 πP

i (a)yi �∑N

i=1 πPi (a′)yi . As usual, costlier actions increase

the expected value of output to the principal.Principal and agent have linear utility over money. The agent evaluates the difference between

the expected value of the contract and the cost of his action for each probability distribution. Foreach a and each π ∈ Π(a), this difference is given by

Eπ [w] − c(a) =N∑

j=1

πjwj − c(a)

Given a contract, the agent chooses an action by computing many expected values (one for eachelement of each belief set).

The principal evaluates the expected value of output minus the expected cost of the contract.For each a, the principal’s expected utility is

EπP (a)[y − w] =N∑

j=1

πPj (a)yj −

N∑j=1

πPj (a)wj

9 We discuss in Section 5 some undesirable effects of this formulation with Knightian uncertainty.

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1154 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

In the standard model, the principal is risk-neutral while the agent is either risk-neutral orrisk-averse. Both parties have the same beliefs, and the same attitude toward uncertainty, butthey might evaluate risk in different ways. In our model, both parties have consistent beliefs andevaluate risk in the same way, but might have different attitudes toward uncertainty. Notice thateven though the agent’s utility function is linear, the agent cares about the distribution of payoffsacross states.

If the agent’s action is observable (and/or verifiable), the contract can depend on it. In keepingwith standard terminology, we denote the cost of this contract by CFB(a), where FB stands forfirst-best. One can immediately see that CFB(a) = w + c(a) for each action a. In other words,the principal can implement a with a contract that pays CFB(a) in every state if action a is takenand −∞ otherwise. A Pareto efficient contract must be constant across states: contracts that donot fully insure the agent are dominated by “flatter” contracts that do not make the agent worseoff and lower the principal’s expected costs.10

The principal and the agent do not value the firm equally. Therefore, one cannot appeal to thestandard way of dealing with moral hazard when the parties have linear utility, which would havethe principal sell the firm to the agent. For a given action, the highest price the agent is willing topay for the firm is the lowest expected value of output minus the cost of taking that action. Thehighest price the agent is willing to pay is thus lower than what the firm is worth to the principal.

3.1. Two implementation rules

A contract implements an action a∗ if it induces the agent to participate and choose thataction. The contract must provide incentives that leave no doubt the agent’s choice is a∗ amongall possible actions.11 How this is done depends on whether or not the inertia assumption holds.

3.1.1. Implementing without inertiaWithout inertia, a contract must satisfy the standard participation and incentive compatibility

conditions, modified to allow for Knightian uncertainty. Participation requires that a∗ is preferredto the reservation utility, while incentive compatibility requires that a∗ is preferred to all otheractions.

Definition 1. An incentive scheme w implements a∗ if:

N∑j=1

πj

(a∗)wj − c

(a∗) � w ∀π

(a∗) ∈ Π

(a∗) (P)

and for each a ∈ A with a = a∗

N∑j=1

πj

(a∗)wj − c

(a∗) �

N∑j=1

πj (a)wj − c(a) ∀π(a∗) ∈ Π

(a∗) and ∀π(a) ∈ Π(a) (IC)

10 Rigotti and Shannon [20] give a general characterization of Pareto optimal allocations with incomplete preferences.11 In [17] we distinguish two notions of incentive compatibility in a general mechanism design framework: optimal andmaximal. A mechanism is optimal incentive compatible if truth telling is preferred to all other reports. A mechanism ismaximal incentive compatible if no report is preferred to truth telling. The notion of incentive compatibility we considerhere corresponds to what we call optimal incentive compatibility in [17].

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For any probability distribution induced by a∗, the expected utility of the contract must beweakly higher than the reservation utility, and weakly higher than the expected utility calculatedaccording to all probability distributions induced by any other action.

The implementation requirements imposed by Definition 1 are restrictive. In particular, therecould be many actions that cannot be implemented.

Proposition 1. If Π(a) ∩ Π(a′) = ∅ for some actions a = a′, then a and a′ cannot both beimplemented.

Proof. Let the two actions be a and a′ and, without loss of generality, assume a > a′. Supposeby way of contradiction that there exists π ∈ Π(a) ∩ Π(a′), and there exists a contract w thatimplements a. Therefore, by (IC),

N∑j=1

πjwj − c(a) �N∑

j=1

πjwj − c(a′)

This implies c(a) � c(a′), and contradicts a > a′. �A possible implication of Proposition 1 is that, when the amount of uncertainty is large, few

actions could be implementable because no action for which the belief set intersects the belief setof a cheaper action can be implemented. For example, as an agent works harder he might gathervaluable information regarding his influence on the productive process. Thus costlier actionsmight lead to a smaller set of beliefs. In such cases, the belief set for a cheaper action will alwaysintersect the belief set of every more expensive action, and only the cheapest action could beimplemented. The addition of inertia, which we discuss next, relaxes incentive constraints andcan enlarge the set of implementable actions.

3.1.2. Implementing with inertiaUnder the inertia assumption, an alternative is chosen only if it is preferred to the status quo.

Hence, actions not comparable to the reservation utility are not chosen. If a∗ and a′ are notcomparable, but a∗ is preferred to the status quo while a′ is not comparable to it, then the inertiaassumption implies a′ is not chosen. Therefore, with inertia a contract does not need to make a∗preferred to all other actions, but just to those that are comparable to the status quo.

Definition 2. An incentive scheme w implements a∗ with inertia in A if

N∑j=1

πj

(a∗)wj − c

(a∗) � w ∀π

(a∗) ∈ Π

(a∗) (P)

and for each a ∈ A with a = a∗, either

N∑j=1

πj

(a∗)wj − c

(a∗) �

N∑j=1

πj (a)wj − c(a) ∀π(a∗) ∈ Π

(a∗) and ∀π(a) ∈ Π(a) (IC)

orN∑

j=1

πj (a)wj − c(a) � w for some π(a) ∈ Π(a) (NC)

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1156 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

(NC) is a non-comparability constraint. It says there exists at least one probability distributioninduced by a such that the corresponding expected utility of the contract is weakly lower than thereservation utility. Inertia weakens the incentive constraints since actions that are not comparableto the outside option are not chosen. Obviously, contracts that satisfy Definition 1 also satisfyDefinition 2. On the other hand, there could be contracts that implement a costlier action withinertia even if belief sets intersect, and therefore inertia enlarges the set of implementable actions.

3.2. The principal’s problem

Given an action, the principal looks for the cheapest contract that implements it. She thendecides which action to implement. As usual, we focus only on the first of these problems. For agiven action a, let H(a) be the (possibly empty) set of incentive schemes that implement a, andHI (a) be the (also possibly empty) set of incentive schemes that implement a with inertia. Wesay w(a) is an optimal incentive scheme to implement a if it is a solution to

minw∈H(a)

N∑j=1

πPj (a)wj (1)

Let W (a) denote the set of solutions to (1). Similarly, we say that w(a) is an optimal incentivescheme to implement a with inertia if it is a solution to

minw∈HI (a)

N∑j=1

πPj (a)wj (2)

Let W I (a) denote the set of solutions to (2).

3.3. Some characteristics of the optimal incentive scheme

Although the agent’s behavior depends on all the probability distributions in his belief set,some are particularly relevant because they determine whether incentive and participation con-straints are satisfied.

For any action a and contract w, let

Π(a;w) := arg minπ(a)∈Π(a)

N∑j=1

πj (a)wj

Given w, any π ∈ Π(a;w) is a probability distribution yielding the minimum expected valuefor the agent when he chooses action a. A contract w satisfies the participation constraint foraction a if and only if Eπ [w] − c(a) � w for every π ∈ Π(a;w). Similarly, given a contract w,action a satisfies the non-comparability constraint if and only if Eπ [w] − c(a) < w for someπ ∈ Π(a;w). For any fixed action a and contract w, let

Π(a;w) := arg maxπ(a)∈Π(a)

N∑j=1

πj (a)wj

Then π ∈ Π(a;w) is a probability yielding the maximum expected value of the contract w forthe agent when he chooses action a. A contract w satisfies the incentive compatibility constraint

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for action a versus a′ if and only if Eπ [w] − c(a) � Eπ [w] − c(a′) for all π ∈ Π(a;w) and allπ ∈ Π(a′;w).

The following proposition extends some standard results about optimal contracts (all proofsare in Appendix A).

Proposition 2. Let a be the action to be implemented.

(i) For any w in W (a) or W I (a), the participation constraint binds when computed accordingto π(a) ∈ Π(a; w); formally,

N∑j=1

πj (a)wj − c(a) = w

for any w ∈ W (a) ∪ W I (a) and π(a) ∈ Π(a; w).(ii) If a is the least costly action, the scheme that pays w + c(1) in all states implements a (with

inertia); that is, w + c(1) ∈ W (1).(iii) If w ∈ W I (a) and a is not the least costly action, then there exists an action less costly than

a such that either (IC) or (NC) binds for this action; formally, if a > 1 there exists an a′ < a

such that either

N∑j=1

πj (a)wj − c(a) =N∑

j=1

πj

(a′)wj − c

(a′)

∀π(a) ∈ Π(a; w) and ∀π(a′) ∈ Π

(a′; w)

or

N∑j=1

πj

(a′)wj − c

(a′) = w ∀π

(a′) ∈ Π

(a′; w)

for any w ∈ W I (a).Similarly, if w ∈ W (a) and a is not the least costly action, then there exists an action lesscostly than a such that (IC) binds for this action.

4. Optimal incentive schemes

In this section, we first consider the case of two actions. Then we generalize the result to manyactions.

The main results depend on two properties of beliefs.

Assumption A-1. For each action a ∈ A, Π(a) is the core of a convex capacity va on N .12 Thatis,

Π(a) = {π ∈ �(N ): π(E) � νa(E) for each E ⊂ N

}(3)

12 A convex capacity on N is a function ν : 2N → [0,1] such that (i) ν(∅) = 0, (ii) ν(N ) = 1, (iii) ∀E,E′ ⊂ N ,E ⊆ E′ implies ν(E) � ν(E′), and (iv) ∀E,E′ ⊂ S, ν(E ∪ E′) � ν(E) + ν(E′) − ν(E ∩ E′).

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1158 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

Fig. 2. Agent’s beliefs with 3 output states.

Geometrically, Assumption A-1 requires that Π(a) is a polyhedron with boundaries deter-mined by the linear inequalities in (3). Fig. 2 displays some examples with three states: ΠA andΠB are cores of convex capacities while ΠC is not.

Some intuition for the content of this assumption might be gleaned from a slightly strongercondition that makes use of probability bounds. That is, for each state i = 1, . . . ,N , take qi, ri ∈[0,1] with qi � ri ; the interval [qi, ri] will be interpreted as bounds on the probability of state i

occurring. Under several consistency conditions,13

Π := {π ∈ �(N ): πi ∈ [qi, ri] ∀i

}is the core of the convex capacity ν defined by

ν(E) = minπ∈Π

π(E) ∀E ⊂ N

A similar restriction is found in much of the work on applications of ambiguity, which fre-quently uses a version of Choquet expected utility [22]. In these models uncertainty aversion anda limited notion of independence generate beliefs represented by the core of a convex capacity.For example, both [18] and [9] assume the beliefs of both parties involved in a moral hazardmodel are represented by the cores of convex capacities.

13 Specifically, the collection {qi , ri : i = 1, . . . ,N} is proper if∑i

qi � 1 �∑i

ri

For a proper collection, the set of probabilities consistent with the associated intervals state by state is nonempty. Thus if{qi , ri : i = 1, . . . ,N} is proper,{

π ∈ �(N ): πi ∈ [qi , ri ] ∀i}

is nonempty. The collection {qi , ri : i = 1, . . . ,N} is reachable if for each i,∑j =i

qj + ri � 1 and∑j =i

rj + qi � 1

This guarantees that

qi = inf{πi : π ∈ �(N ), πj ∈ [qj , rj ] ∀j

}and ri = sup

{πi : π ∈ �(N ), πj ∈ [qj , rj ] ∀j

}for each state i. If {qi , ri : i = 1, . . . ,N} is proper and reachable, then the set Π consistent with these bounds is the coreof a convex capacity, in particular ν(E) = minπ∈Π π(E) ∀E ⊂ N . For example, see [6].

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1159

For simplicity we have chosen to state Assumption A-1 directly as a technical condition onthe belief sets. We could give an alternative formulation based on conditions on the agent’s un-derlying preferences, however. That is, for each a ∈ A, let �a denote the agent’s strict preferenceorder over state-contingent monetary payoffs RN , conditional on choosing the action a. Givenour assumptions, for any pair w,y ∈ RN ,

w �a y ⇐⇒ Eπ [w] − c(a) > Eπ [y] − c(a) ∀π ∈ Π(a)

Assumption A-1 can be rephrased as a condition on �a by making use of a natural extension ofthe notion of subjective beliefs in [21].

Assumption A-2. For each action a ∈ A, πP (a) is an element of the relative interior of Π(a).14

In addition to the relative imprecision condition we maintain by assuming πP (a) ∈ Π(a) foreach a, Assumption A-2 requires that the agent perceives sufficiently rich uncertainty so thatΠ(a) has a nonempty relative interior for each action a, and refines the restriction on principal’sbeliefs to rule out cases where the principal’s belief is an extreme element of the agent’s set ofbeliefs.

4.1. Optimal incentive schemes with two actions

Assume there are only two actions, H and L, with c(H) > c(L). We interpret these as highand low effort respectively. By Proposition 2, the principal can implement low effort at the firstbest cost with a constant contract. Our main result shows that a contract implementing high effortcan be optimal only if it takes two values.

Proposition 3. Suppose Assumptions A-1 and A-2 hold, and the high action H is implementable(implementable with inertia). Then, the unique optimal incentive scheme to implement H (withinertia) divides the N states into two subsets and is constant on each subset.

This result suggests Knightian uncertainty could explain the emergence of simple contracts.Without it, our setup reduces to the standard principal-agent model with linear utilities. In thatcase, both parties are risk-neutral and a continuum of contracts is optimal. These include simpletwo-valued contracts and many complicated ones. The simplest in this class is the contract thatcorresponds to selling the firm to the agent, hence avoiding incentive problems altogether. Inour setup, because of Knightian uncertainty the parties cannot avoid the incentive problem evenwhen utility functions are linear. Furthermore, a two-tier contract is the unique optimal contractwith Knightian uncertainty. Adding Knightian uncertainty to the standard model selects a unique,simple contract.

Our model predicts simple contracts when the standard model with risk-averse agent doesnot. Although the standard model makes a different assumption on the curvature of the agent’sutility function, it shares some behavioral characteristics with the particular form of Knightianuncertainty we assume, since the agent’s better-than sets are not half-spaces but proper cones.With Knightian uncertainty, thinking of an agent as risk-neutral when his utility function is linearseems inappropriate since he cares about the distribution of payoffs across states.

14 Here we mean interior relative to �(N ), that is, the points π ∈ Π(a) such that there exists an open set O ⊂ RN withπ ∈ O ∩ �(N ) ⊂ Π(a).

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1160 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

Notice that the inertia assumption plays no role in Proposition 3. In Appendix A, we providea detailed proof for the inertia case, but the other case can be proved very similarly. Intuitively,the result follows from a simple consequence our assumptions. For any state, there is at least onebelief of the agent that assigns lower probability than the principal to that state. Therefore, for anystate there is a belief that makes the principal more optimistic about the likelihood of that state.Because all beliefs matter, the optimal contract must “accommodate” this probability and doingso is costly to the principal. Since this is true for all states, and utility is linear, the principal canalways benefit by using a “flatter” contract. Eventually, this process must stop because a contractmust have enough variation to provide incentives.

Next we show that Proposition 3 is robust to the introduction of a small degree of risk aver-sion. Consider the generalization of our basic model in which the agent’s utility function overcertain monetary outcomes is a strictly increasing and strictly concave function u : R → R thatis twice continuously differentiable, with u′(x) > 0, u′′(x) < 0 for all x ∈ R. Let vM denote thehighest utility level that the agent can receive, i.e. vM = u(V∗), where V∗ denotes the highestprofit generated by any action. Let h : R → R denote the inverse of u. Since h is strictly increas-ing and convex, we can represent it as the sum h(u) = u + g(u), where g : R → R is convexand continuously differentiable. Our main result remains valid provided g′ is sufficiently small.Because the derivation of the requisite bound is somewhat complicated, we defer the proof toAppendix A.

Proposition 4. Suppose Assumptions A-1 and A-2 hold, and the high action H is implementable(implementable with inertia). Then there exists b > 0 such that if g′(vM) < b, then the uniqueoptimal incentive scheme to implement H (with inertia) divides the N states into two subsets andis constant on each subset.

The idea is that when the curvature of the utility function is small, Knightian uncertaintydominates, and the intuition behind Proposition 3 is powerful enough to overcome the usualarguments that make the optimal contract depend on the likelihood ratios of different states acrossdifferent actions.

4.2. Optimal incentive schemes with many actions

We now provide conditions to extend the previous results to many actions. Simple reasoningshows that the main result is easily extended to more actions without additional restrictions; itimplies that an optimal contract takes as many values as there are actions. The proof of Proposi-tion 3 shows that reducing the number of states on which a contract depends is profitable for theprincipal. Clearly, this becomes impossible when all constraints are binding. In the case of twoactions, this implies the contract has two values. If there are M possible actions, there will be atmost M binding constraints, and the optimal contract must be at most M-valued.

Under additional restrictions, we can show that the optimal contract must be two-valued evenwhen there are many actions. These conditions reduce the many-action case to the two-actioncase. The exercise parallels what is done to obtain monotonicity (in output) of the optimal con-tract in the standard model, and uses analogous assumptions.

Assumption A-3. For each action a, every selection in{πe: A → �(N )

∣∣ πe(a) is an extreme point of Π(a) for each a}

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1161

satisfies the monotone likelihood ratio property and concavity of the distribution function. Thatis, if πe : A → �(N ) is a selection such that πe(a) is an extreme point of Π(a) for each a, then

(i) (MLR) for any two actions a′, a, c(a) > c(a′) implies thatπe

i (a′)πe

i (a)is decreasing in i.

(ii) (CDFC) for any three actions a′′, a′, a such that c(a) = λc(a′) + (1 − λ)c(a′′) for someλ ∈ [0,1], ∑j

i=1 πei (a) � λ

∑j

i=1 πei (a′) + (1 − λ)

∑j

i=1 πei (a′′) for all j .

The first part of this assumption says that the standard monotone likelihood ratio assumptionholds for any extreme point of the set of probability distributions induced by each action. Thesecond part extends concavity of the distribution function along similar lines by also requiringthat it holds for any extreme point of the set of probability distribution induced by each action.Without uncertainty, these are pretty standard assumptions in the moral hazard literature. Withuncertainty, all we do is extend them in the most obvious fashion so that they hold for the sets ofprobability distribution of our previous assumptions.

Proposition 5. Suppose Assumptions A-1, A-2 and A-3 hold. Then for any action a∗ > 1, theoptimal contract to implement a∗ (with inertia) has a two-wage structure.

Interestingly, a sufficient condition to reduce the multi-action case to the two-action case isa generalized version of the requirement one needs in the standard model to obtain monotonic-ity in output of the optimal contract. Notice that, a fortiori, this same condition would inducemonotonicity of the contract in our model.

5. The primitive model

The standard principal-agent model is formalized by using probability distributions that de-pend on actions. This is a shortcut for the formal model considered by the early literature inwhich actions determine output jointly with the state of the world while probabilities over statesare given. As noted in [15], the parameterized distribution model lacks an axiomatic founda-tion since in both Savage and Anscombe–Aumann probability distributions over states do notdepend on choices. Without Knightian uncertainty, any primitive model can be written in theparameterized distribution formulation but not vice versa (see [13] for a discussion). Hence as-sumptions made directly on the parameterized distribution model may not have a counterpartin any primitive model and thus may not be consistent with first principles. In this section, weformalize the primitive model with Knightian uncertainty, and show that our main assumptionscan be deduced from similar assumptions on preferences in that model. In the process, we showhow action-based beliefs over outcomes are derived from a fixed set of beliefs over states andtherefore establish that the parameterized distribution formulation can also be thought of as ashortcut for the primitive model in the presence of Knightian uncertainty.

Let S be the state space, with elements s = 1, . . . , S. Output is determined by a produc-tion function that depends on effort as well as stochastic factors. For each action a ∈ A,y(a) :S → R denotes the output that results from the action a. For each action a, a′, we as-sume suppy(a) = suppy(a′) := {y1, . . . , yN }; different output levels across actions correspondto different permutations of the state space, so that observing a particular output level does notreveal the agent’s action.

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1162 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

If the output is y, the principal’s payoff is y − w. The agent’s payoff is w − c(a), where asabove c : A → R denotes the agent’s disutility of effort. The wage depends on output alone andnot on the state. Since the agent knows which action he has taken, he also knows which state hasoccurred. The principal, on the other hand, does not know the realized state of the world since hecan only observe realized output. We assume that the agent’s beliefs over S are given by a closedand convex set Π ⊂ �(S), and that the principal’s belief is πP ∈ �(S).

Our notions of implementation can be reformulated as follows.

Definition 3. An incentive scheme w implements a∗ if:

S∑s=1

πsw(ys

(a∗)) − c

(a∗) � w ∀π ∈ Π (P)

and for each a ∈ A with a = a∗

S∑s=1

πsw(ys

(a∗)) − c

(a∗) �

S∑s=1

πsw(ys(a)

) − c(a) ∀π ∈ Π (IC)

Definition 4. An incentive scheme w implements a∗ with inertia in A if

S∑s=1

πsw(ys

(a∗)) − c

(a∗) � w ∀π ∈ Π (P)

and for each a ∈ A with a = a∗, either

S∑s=1

πsw(ys

(a∗)) − c

(a∗) �

S∑s=1

πsw(ys(a)

) − c(a) ∀π ∈ Π (IC)

orS∑

s=1

πsw(ys(a)

) − c(a) � w for some π ∈ Π (NC)

To relate this model to the parameterized distribution model, given π ∈ Π and a ∈ A, defineπ(a) ∈ �(N ) by

πi(a) =∑

{s∈S : ys(a)=yi }πs

For each a, set Π(a) := {π ∈ �(N ): π = π(a) for some π ∈ Π}. Notice that Π(a) is closedand convex for each a. This means assumptions about the set Π can easily be translated intoassumptions about Π(a). In particular, we show that Assumptions A-1 and A-2 can be derivedfrom analogous assumptions on Π (the beliefs over the primitive state space).

For each E ⊂ {1, . . . ,N} and each a, let E(a) := {s ∈ S : ys(a) ∈ E}. Then notice that π ∈Π(a) ⇔ ∃π ∈ Π such that π(E) = π(E(a)) for each E ⊂ {1, . . . ,N}.

Proposition 6. Let ν : 2S → [0,1] be a convex capacity on S and let Π be the core of ν. Foreach a ∈ A define νa : 2N → [0,1] by νa(E) = ν(E(a)) for each E ⊂ {1, . . . ,N}. Then for eacha ∈ A, νa is a convex capacity on {1, . . . ,N} with core Π(a).

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1163

We say that a subset B ⊂ S is unambiguous if π(B) = π ′(B) for all π,π ′ ∈ Π . While theempty set and the entire state space S are always unambiguous, every nonempty, proper subsetof S is ambiguous under the assumption that Π has nonempty relative interior. Here we assumethat Π has nonempty relative interior, and that the principal’s unique prior πP ∈ �(S) is anelement of rel intΠ . Under these assumptions, together with the assumption that the support ofstochastic output y(a) is the same for all actions a, no action eliminates ambiguity. That is, Π(a)

also has a nonempty relative interior, and πP (a) ∈ rel intΠ(a) for each action a.

Proposition 7. If πP ∈ rel intΠ , then πP (a) ∈ rel intΠ(a) for each action a.

These two results allow us to identify assumptions in this primitive model that will suffice toensure that optimal contracts robust to uncertainty will be simple, as in the previous section.

Assumption A-4. Π is the core of a convex capacity ν on S .

Assumption A-5. πP is an element of the relative interior of Π .

Proposition 8. Suppose Assumptions A-4 and A-5 hold, and the high action H is implementable(implementable with inertia) in {L,H }. Then, the unique optimal incentive scheme to imple-ment H (with inertia) divides the S states into two subsets and is constant on each subset.

6. Conclusions

We study a principal-agent model with Knightian uncertainty and asymmetric confidence inbeliefs. In this model, optimal contracts must be robust to the agent’s imprecise beliefs about thestochastic relationship between effort and output, as well as reflect the fact that the principal ismore confident than the agent in evaluating that relationship. Our main result shows that optimalcontracts must be two-valued, and are therefore simpler than the contracts that frequently obtainin the standard principal-agent model. This result also allows for the presence of a small amountof curvature in the agent’s utility function. Knightian uncertainty thus introduces a force thatpushes incentive schemes to be as flat as possible, while still providing incentives. This forcecan be sufficient to prevail over the usual motives that make contracts highly dependent on finedetails of beliefs.

Appendix A

Proof of Proposition 2. We prove each statement separately.Proof of (i): Suppose not. Then w(a) is optimal and

∑Nj=1 πj (a)wj (a) − c(a) > w. Reduce

the payment in each state by ε > 0; that is, for all j , let wj = waj − ε. For ε small enough, wj

implements a according to both definitions because it satisfies (P), (IC), and (NC). Furthermore,∑Nj=1 πP

j (a)wj <∑N

j=1 πPj (a)wj (a) − ε, contradicting the optimality of w(a).

Proof of (ii): Because πP (1) ∈ Π(1), for any payment scheme w and any π(1) ∈ Π(1;w),by definition

∑Nj=1 πj (1)wj �

∑Nj=1 πP

j (1)wj . The constant contract wj = w + c(1) for all j

is feasible: it satisfies (P), and it satisfies (IC) because alternative actions are more costly for theagent. It is a solution to (1) because w + c1 = ∑N

πj (1)wj (1) = ∑NπP (1)wj (1).

j=1 j=1 j
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1164 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

Proof of (iii): Suppose the claim does not hold. That is, w(a) is optimal and for each a′ < a,either

N∑j=1

πj (a)wj (a) − c(a) >

N∑j=1

πj

(a′)wj (a) − c

(a′)

for all π(a) ∈ Π(a; w) and π(a′) ∈ Π(a′; w), or

N∑j=1

πj (a′)wj (a) − c(a′) < w

for some π(a′) ∈ Π(a′). The latter inequality implies∑N

j=1 πj (a′)wj (a) − c(a′) < w for any

π(a′) ∈ Π(a′; w). Because none of the respective constraints binds, w(a) is a solution for aproblem like (1) where all actions like a′ have been dropped from the constraints. In this newproblem, a is the least costly action and a contract that pays w + c(a) in all states is optimal.That is, wj = w+c(a) for each j . Thus, w+c(a) = ∑N

j=1 πj (a)wj (a) = ∑Nj=1 πj (a

′)wj (a) =∑Nj=1 πj (a

′)wj (a). Hence c(a′) > c(a), contradicting a′ < a. �Proofs for Section 4

We will use the following results adapted from [7].

Lemma 1. Let v be a convex capacity on S with core Π , and let Πe be the set of extreme pointsof Π . Then, for any N -vector z such that z1 � z2 � · · · � zN ,

minπ∈Π

N∑j=1

πjzj

is attained at π ∈ Πe such that∑N

s=j πs = ∑Ns=j v({s}) for each j = 2, . . . ,N . In particular, if

z and z′ are two N -vectors such that z1 � · · · � zN and z′1 � · · · � z′

N , then

arg minπ∈Π

N∑j=1

πjzj = arg minπ∈Π

N∑j=1

πjz′j

The first result follows from [7, Propositions 10 and 13]. The second follows from the first,which shows that the set of minimizing distributions depends only on the order of the elementsof z and not on the values of the components.

When Assumption A-1 holds, the lemma provides a characterization of the probability distri-butions that, among a set, yield the smallest and largest expected values of a fixed contract. Inparticular, this lemma can be used to show that for each action a, the smallest expected value fora given z is attained at an extreme point of Π(a) such that πK(a) < πP

K (a) and π1(a) � πP1 (a),

where K is an index at which the maximum value of zk is attained. Moreover, when two incentiveschemes are “ordered” in the same way, their minimum or maximum expectations are attainedusing the same distributions.

Proof of Proposition 3. The main step is to show, by contradiction, than an optimal contractcannot be contingent on more than two subsets of output levels. Define a partition of the state

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1165

space such that each element in this partition corresponds to different payments in the optimalcontract w. Let k be an element of this partition; we say k is an event. Any probability distributionover the original space defines a probability distribution over this partition. Label events so thatK corresponds to the largest wk , K − 1 to the second largest, and so on until event 1 correspondsto the smallest value of wk . By construction, K is the number of events the optimal contract iscontingent upon, and w1 < w2 < · · · < wK . We need to show K equals 2.

By Proposition 2, w must satisfyK∑

k=1

πk(H)wk = w + c(H) (4)

for any π(H) ∈ Π(H ; w). We claim w must also satisfy

K∑k=1

πk(L)wk = w + c(L) (5)

for any π(L) ∈ Π(L; w). Suppose not. Then,∑K

k=1 πk(L)wk < w + c(L). Let

w =(

w1 − επK(H)

π1(H), w2, . . . , wK−1, wK + ε

)where ε is positive and small enough so that the ranking of the payments for w and w is thesame. By Lemma 1, π(H) and π(L) minimize the expected value of w for the agent. For anydistribution π , the expected values of w and w are related by the following:

K∑k=1

πkwk −K∑

k=1

πkwk = π1πK(H) − πKπ1(H)

π1(H)ε (6)

π = π(H) implies the right-hand side of (6) is equal to 0; thus, w satisfies (4). For some ε

close enough to zero and π = π(L) the right-hand side of (6) is very small; thus w satisfies(NC) because w satisfies it strictly. By Lemma 1 and Assumption A-1, πK(H) < πP

K (H) andπ1(H) � πP

1 (H); thus, the right-hand side of (6) is negative when π = πP (H). Summarizing,w is feasible and cheaper than w, contradicting the optimality of w. Hence (5) must hold for w

to be an optimum.We claim K must be strictly larger than 1. Suppose not, i.e., K = 1. If this is the case, all

payments are the same, and the left-hand sides of Eqs. (4) and (5) are the same. Thus, we havew + c(L) = w + c(H), contradicting c(H) > c(L).

We claim K is not larger than 2. Suppose not. Then w is optimal, so satisfies Eqs. (4) and (5),and K > 2. Eqs. (4) and (5) constitute a system of two equations which can be solved for wK

and some wk′ , yielding:

wK = πk′(L)(w + c(H)) − πk′(H)(w + c(L))

πk′(L)πK(H) − πK(L)πk′(H)

+∑

k =K,k′

πk′(H)πk(L) − πk(H)πk′(L)

πk′(L)πK(H) − πK(L)πk′(H)wk (7)

wk′ = πK(H)(w + c(L)) − πK(L)(w + c(H))

πk′(L)πK(H) − πK(L)πk′(H)

+∑

πK(L)πk(H) − πK(H)πk(L)

πk′(L)πK(H) − πK(L)πk′(H)wk (8)

k =K,k

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1166 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

These are well defined unless

πK(L)πk′(H) − πk′(L)πK(H) = 0 for all k′ = K (9)

In that case:

0 = πK(L)

K−1∑k=1

πk(H) − πK(H)

K−1∑k=1

πk(L)

= πK(L)(1 − πK(H)

) − πK(H)(1 − πK(L)

)= πK(L) − πK(H)

⇒ πK(H) = πK(L)

Using this result in (9):

πK(H)(πk′(H) − πk′(L)

) = 0

Thus, either πK(H) = πK(L) = 0, or πk′(H) = πk′(L) for all k′ = K . If πK(H) = πK(L) = 0,w cannot be optimal because it makes the largest payment in a state that does not affect the con-straints and, by Assumption A-2, has positive probability for the principal. If πk′(H) = πk′(L)

for all k′ = K , then π(L) = π(H). In this case,∑K

k=1 πk(H)wk = ∑Kk=1 πk(L)wk and c(H) =

c(L), a contradiction.Using Eqs. (7) and (8), we can write the expected cost of the optimal incentive scheme as

follows:K∑

k=1

πPk (H)wk = α +

∑k =K,k′

βkwk

where

α = πPK (H)πk′(L) − πP

k′ (H)πK(L) + πPk′ (H)πK(H) − πP

K (H)πk′(H)

πk′(L)πK(H) − πK(L)πk′(H)w

+ πPK (H)πk′(L) − πP

k′ (H)πK(L)

πk′(L)πK(H) − πK(L)πk′(H)c(H)

+ πPk′ (H)πK(H) − πP

K (H)πk′(H)

πk′(L)πK(H) − πK(L)πk′(H)c(L)

and

βk = πPk (H) − πk(H)

πPK (H)πk′(L) − πP

k′ (H)πK(L)

πk′(L)πK(H) − πK(L)πk′(H)

− πk(L)πP

k′ (H)πK(H) − πPK (H)πk′(H)

πk′(L)πK(H) − πK(L)πk′(H)(10)

We claim that βk′′ = 0 for some k′′ = K,k′. Suppose not. Then, βk = 0 for each k = K,k′.Using Eq. (10),

πPk (H)

(πk′(L)πK(H) − πK(L)πk′(H)

)= πk(H)

(πP (H)πk′(L) − πP′ (H)πK(L)

) + πk(L)(πP′ (H)πK(H) − πP (H)πk′(H)

)

K k k K
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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1167

Summing over k, rearranging, and solving for πPk′ (H):

πPk′ (H) = −πk′(L)πK(H) − πK(L)πk′(H) + πP

K (H)(πk′(H) − πk′(L))

πK(L) − πK(H)

This implies:

πPK (H)πk′(L) − πP

k′ (H)πK(L)

πk′(L)πK(H) − πK(L)πk′(H)= πK(L) − πP

K (H)

πK(L) − πK(H)

and

πPk′ (H)πK(H) − πP

K (H)πk′(H)

πk′(L)πK(H) − πK(L)πk′(H)= πP

K (H) − πK(H)

πK(L) − πK(H)

We know that πK(H) < πPK (H). Thus,

πK(L)−πPK (H)

πK(L)−πK(H)< 1 and

πPK (H)−πK(H)

πK(L)−πK(H)> 0. Hence:

K∑k=1

πPk (H)wk = α = w + c(H) + πP

K (H) − πK(H)

πK(L) − πK(H)

(c(L) − c(H)

)< w + c(H)

a contradiction.Because βk′′ = 0 for some k′′ = K,k′, we find a feasible contract which is cheaper than w.

Let w be defined as follows:

wk = wk when k = K,k′, k′′

wK = wK + πk′(H)πk′′(L) − πk′′(H)πk′(L)

πk′(L)πK(H) − πK(L)πk′(H)ε

wk′ = wk′ + πK(L)πk′′(H) − πK(H)πk′′(L)

πk′(L)πK(H) − πK(L)πk′(H)ε

wk′′ = wk′′ + ε

where

|ε| < min{wk′′ − wk′′−1, wk′′+1 − wk′′ , wK − wK−1, wk′ − wk′−1, wk′+1 − wk′ }By construction, the payments in w and w are ranked in the same order. Lemma 1 applies, andπ(H) and π(L) yield the minimum expected values of w under actions H and L. Moreover,

0 = πK(L)πk′(H)πk′′(L) − πk′′(H)πk′(L)

πk′(L)πK(H) − πK(L)πk′(H)

+ πk′(L)πK(L)πk′′(H) − πK(H)πk′′(L)

πk′(L)πK(H) − πK(L)πk′(H)+ πk′′(L)

0 = πK(L)πk′(H)πk′′(L) − πk′′(H)πk′(L)

πk′(L)πK(H) − πK(L)πk′(H)

+ πk′(L)πK(L)πk′′(H) − πK(H)πk′′(L)

πk′(L)πK(H) − πK(L)πk′(H)+ πk′′(L)

Hence, w is feasible because w is. The expected cost of w is given by

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1168 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

K∑k=1

πPk (H)wk = α +

∑k =K,k′

βkwk

= α +∑

k =K,k′βkwk + βk′′ε

Thus, we can choose ε > 0 whenever βk′′ < 0 and ε < 0 whenever βk′′ > 0. In either case, w isfeasible and cheaper than w, contradicting the optimality of w. Summarizing, if an optimal con-tract is contingent on K > 2 events, we can find a feasible contract which is cheaper. Therefore,because we already proved K < 2 is impossible, a contract can be optimal only if K = 2. �Proof of Proposition 4. To simplify notation, throughout we use πP in place of πP (H).

First, set

S :={(

A,B,C,π1,π2): A,B,C ⊂ N are pairwise disjoint,

π1 ∈ Π(H), π2 ∈ Π(L) are extreme points,

π1(A)π2(B) − π1(B)π2(A) = 0∣∣∣∣πP (B)π1(C)π2(A) − π1(A)π2(C)

π1(A)π2(B) − π1(B)π2(A)

∣∣∣∣+

∣∣∣∣πP (A)π1(B)π2(C) − π1(C)π2(B)

π1(A)π2(B) − π1(B)π2(A)

∣∣∣∣ = 0∣∣∣∣πP (C) + πP (B)π1(C)π2(A) − π1(A)π2(C)

π1(A)π2(B) − π1(B)π2(A)

+ πP (A)π1(B)π2(C) − π1(C)π2(B)

π1(A)π2(B) − π1(B)π2(A)

∣∣∣∣ = 0

}Notice that S is finite; we argue below that S is also nonempty.For any pairwise disjoint events A,B,C ⊂ N and any extreme points π1 ∈ Π(H) and π2 ∈

Π(L) with (A,B,C,π1,π2) ∈ S, set

R(A,B,C,π1,π2):=

|πP (C) + πP (B)π1(C)π2(A)−π1(A)π2(C)

π1(A)π2(B)−π1(B)π2(A)+ πP (A)

π1(B)π2(C)−π1(C)π2(B)

π1(A)π2(B)−π1(B)π2(A)|

|πP (B)π1(C)π2(A)−π1(A)π2(C)

π1(A)π2(B)−π1(B)π2(A)| + |πP (A)

π1(B)π2(C)−π1(C)π2(B)

π1(A)π2(B)−π1(B)π2(A)|

By definition, R is well defined and strictly positive.Finally, set

b := min(A,B,C,π1,π2)∈S

R(A,B,C,π1,π2)

Since S is finite, b is positive.Arguments similar to the ones used in the first part of the proof of Proposition 2, with each

payment variable wk replaced by the utility variable vk := u(wk), can be used to establish that:(i) (P) must hold with equality for all π ∈ Π(H ;v), and (ii) (NC) must hold with equality forall π ∈ Π(H ;v) and π ∈ Π(L;v). Thus we can restrict the search for optimal contracts in thesubset F of Rm defined by the set of vectors v for which (P) and (NC) hold with equality.

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1169

Suppose that v1, . . . , vK are the agent’s utility levels corresponding to an optimal contractwith K levels w1 < · · · < wK, i.e. vk := u(wk), and suppose by way of contradiction that K > 2.

Thus v1, . . . , vK solves the following program:

minv1,...,vK

K∑k=1

πPk h(vk)

subject to (P) and (NC).

Claim 1. For any v ∈ F there exists j ∈ {1, . . . ,K} such that (NC) and (P) can be solved for(vj , vK), i.e.

vj = πK(w0 + cL) − πK(w0 + cH )

πKπj − πjπK

+∑

k =j,K

πkπK − πKπk

πKπj − πjπK

vk = αj +∑

k =j,K

σjkvk

vK = πj (w0 + cL) − πj (w0 + cH )

πjπK − πKπj

+∑

k =j,K

πjπk − πkπj

πKπj − πjπK

vk = αK +∑

k =j,K

σKkvk

(11)

where πKπj − πjπK = 0, and π ∈ Π(L;v),π ∈ Π(H ;v) are extreme points.

Proof. See the part of the proof of Proposition 2 which begins with “These are well definedunless” immediately before formula (9) (here we have replaced k′ with j ). �

Substituting the expressions in (11) into the objective yields

K∑i=1

πPk h(vi) =

∑k =j,K

πPk h(vk) + πP

j h

(αj +

∑k =j,K

σjkvk

︸ ︷︷ ︸vj

)+ πP

K h

(αK +

∑k =j,K

σKkvk

︸ ︷︷ ︸vK

)

Since the given contract v1, . . . , vK is optimal it must minimize the last expression, hence satisfythe first order condition (FOC):

∀k = j,K: 0 = πPk h′(vk) + πP

j σjkh′(vj ) + πP

K σKkh′(vK)

= [πP

k + πPj σjk + πP

KσKk

]h′(vk)︸ ︷︷ ︸

T1

+ πPj σjk

[h′(vj ) − h′(vk)

] + πPKσKk

[h′(vK) − h′(vk)

]︸ ︷︷ ︸

T2

Claim 2. Since πP = π, for any v ∈ F there exists k∗ = j,K such that the first term in squarebrackets

T1 = πPk + πP

j σjk + πPK σKk

is not zero.

Proof. See the part of the proof of Proposition 3 which establishes that βk′′ = 0 for somek′′ = k′,K . �

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1170 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

Remark. Given Claim 2, the FOC is violated (for vk∗ ) if h′ is constant (i.e. the agent’s utilityfunction is linear); or if h′ does not vary too much, so that

|T1| > |T2|The remainder of the argument verifies that this holds provided g′(vM) < b.

Note that by Lemma 1, the two sets of critical probabilities Π(H ;w) and Π(L;w) do notchange if the order of the payments in the contract does not change.

By Claims 1 and 2 we can conclude that for any v ∈ F there exists k∗(v) such that the firstterm in square brackets (T1) is not zero.

Then the FOC can be written as

∀k = j,K: 0 = [πP

k + πPj σjk + πP

K σKk

][1 + g′(vk)

] + πPj σjk

[g′(vj ) − g′(vk)

]+ πP

K σKk

[g′(vK) − g′(vk)

]Recall that h is unique up to affine transformations. Thus without loss of generality we can setg′(u) = 0, where u denotes the agent’s reservation utility, i.e. u := u(w). This together with theconvexity of g implies g′(u) � 0 for all u > u(w); and in particular g′(vk) � 0. Thus it sufficesto show that the first term in square brackets is larger in absolute value than the term on the lastline (T2) in absolute value.

Since by assumption g′(vM) < b, we have∣∣πPj σjk

[g′(vj ) − g′(vk)

] + πPK σKk

[g′(vK) − g′(vk)

]∣∣�

∣∣πPj σjk

[g′(vj ) − g′(vk)

]∣∣ + ∣∣πPKσKk

[g′(vK) − g′(vk)

]∣∣ (triangle inequality)

= ∣∣πPj σjk

∣∣∣∣g′(vj ) − g′(vk)∣∣ + ∣∣πP

KσKk

∣∣∣∣g′(vK) − g′(vk)∣∣ |xy| = |x||y|

�[∣∣πP

j σjk

∣∣ + ∣∣πPK σKk

∣∣][g′(vM) − g′(u)]

(convexity of g)

= [∣∣πPj σjk

∣∣ + ∣∣πPK σKk

∣∣]g′(vM)(g′(u) = 0

)<

[∣∣πPj σjk

∣∣ + ∣∣πPK σKk

∣∣] |πPk + πP

j σjk + πPK σKk|

|πPj σjk| + |πP

K σKk|(g′(vM) < b

)= ∣∣πP

k + πPj σjk + πP

K σKk

∣∣Since this is a contradiction, the contract could not be optimal and the result is established. �Proof of Proposition 5. Again we prove the result for the case of implementation with inertia;the other case is analogous. Let w be an optimal contract that implements a∗ with inertia. With-out loss of generality, label payments so that wN corresponds to the highest, wN−1 the secondhighest, and so on. We claim there exists only one action a′ = a∗ for which the incentive com-patibility constraint binds, and a′ < a∗. Suppose not. Then, there exist two actions a′ and a′′different from a∗ such that

N∑j=1

πj

(a′′)wj − c

(a′′) =

N∑j=1

πj

(a′)wj − c

(a′) =

N∑j=1

πj

(a∗)wj − c

(a∗) = w

for any π(a∗) ∈ Π(a∗; w), π(a′) ∈ Π(a′; w), and π(a′′) ∈ Π(a′′; w). By construction π(a∗),π(a′), and π(a′′) can be taken to be extreme points of the corresponding belief sets. From hereon, following the argument in [11] establishes the claim.

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G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172 1171

If only the constraint relative to one action a′ ∗ binds, w must also be optimal in a problemwhere all other actions a = a′ ∗ are dropped from the constraints. Therefore, Proposition 3 appliesto that problem and the optimal contract has a two-wage structure. �Proofs for Section 5

Proof of Proposition 6. Fix a ∈ A. Let E,F ⊂ {1, . . . ,N}. Then

νa(E) + νa(F ) = ν(E(a)

) + ν(F(a)

)� ν

(E(a) ∩ F(a)

) + ν(E(a) ∪ F(a)

)= ν

((E ∩ F)(a)

) + ν((E ∪ F)(a)

)= νa(E ∩ F) + νa(E ∪ F)

Thus νa is a convex capacity. Moreover, if π ∈ Π(a), then π = π(a) for some π ∈ Π . Foreach π ∈ Π(a), π(E) = π(E(a)) � ν(E(a)) = νa(E). Thus Π(a) is a subset of the core of νa .Since Π(a) is closed and convex, it suffices to show that Π(a) contains all of the “marginalcontribution” vectors for νa , that is, any vector π of the form πj = νa({σ(1), . . . , σ (j)}) −νa({σ(1), . . . , σ (j − 1)}) where σ is a permutation on {1, . . . ,N}. Without loss of generality,consider the identity permutation and corresponding vector π in which πj = νa({1, . . . , j}) −νa({1, . . . , j − 1}) for each j . Thus for each j , πj = ν({y1, . . . , yj }(a)) − ν({y1, . . . , yj−1}(a)).

For each j , set {sj

1 , . . . , sjkj

} := {yj }(a) = {y1, . . . , yj }(a) \ {y1, . . . , yj−1}(a). Define π ∈ �(S)

as follows. Set

π(s1

1

) = ν({

s11

})and for each k = 2, . . . , k1, set

π(s1k

) = ν({

s11 , . . . , s1

k

}) − ν({

s11 , . . . , s1

k−1

})For j = 2, . . . ,N , similarly define

π(sj

1

) = ν({y1, . . . , yj−1}(a) ∪ {

sj

1

}) − ν({y1, . . . , yj−1}(a)

)and for k = 2, . . . , kj ,

π(sjk

) = ν({y1, . . . , yj−1}(a) ∪ {

sj

1 , . . . , sjk

}) − ν({y1, . . . , yj−1}(a) ∪ {

sj

1 , . . . , sj

k−1

})Then π is an element of the core of ν, since it is the marginal contribution vector correspondingto the permutation (s1

1 , . . . , s1k1

, . . . , sN1 , . . . , sN

kN). Thus π ∈ Π . By construction, for each j ,

π({yj }(a)

) =kj∑

k=1

π(sjk

) = ν({y1, . . . , yj }(a)

) − ν({y1, . . . , yj−1}(a)

) = πj

Thus π ∈ Π(a), and the claim is established. �Proof of Proposition 7. Fix an action a. For each j = 1, . . . ,N , {yj }(a) ⊂ S is a nonemptyand proper subset of S , since y(a) has support {y1, . . . , yN }. Thus for a nonempty, proper subsetE ⊂ {1, . . . ,N}, E(a) ⊂ S must also be nonempty and proper. By assumption,

min π(E(a)

)< πP

(E(a)

)< maxπ

(E(a)

)

π∈Π π∈Π
Page 25: Knightian uncertainty and moral hazardcshannon/wp/agent-jet.pdfKnightian uncertainty and moral hazard ... Ben Polak, the associate editor, and referees for thoughtful comments. * Corresponding

1172 G. Lopomo et al. / Journal of Economic Theory 146 (2011) 1148–1172

From this we conclude

minπ∈Π(a)

π(E) < πPj (a)(E) < max

π∈Π(a)π(E)

Thus the claim is established. �References

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